The problem you have is one of 'thermal diffusivity'. For a simple
one-dimensional case, the problem can be worked out with calculus. You need
to know two parameters of the copper plate, it's heat capacity per unit
volume in terms of Joules/cm^3 (don't confuse this with the heat capacity
per unit mass. If you have heat capacity per unit mass, multiply it by the
density of the substance to get heat capacity per unit volume), and the
second parameter is thermal conductivity (Watts/m-C).
The key here is that it isn't a 'steady-state' problem, but a transient one.
For a simple case, assume the copper at one edge is at 100C at time 0
seconds, and the entire rest of the plate is at 20C at the same time 0. So
heat begins to conduct from the hot edge toward material a short distance
away at the rate of....
Q = conductivity*area*(temperature gradient)
(remember the area is that area perpendicular to the heat flow, so it is the
'thin' edge of your plate, not the area of the plate as you look at it from
the side). The temperature gradient is simply the difference in temperature
divided by the distance between two points (dT/dX).
How fast that amount of heat raises the temperature of the cooler material
is a function
dT/dt = Q/ (area*dX*volumetric heat capacity)
Substituting for Q, we get...
dT/dt = (conductivity / volumetric heat capacity)*d^2T / dX^2
(sorry about this, but dT on the left is the change in temperature with
respect to time, while dT on the right is the difference in temperature
between two points along the plate, not the same thing, so don't try to
combine them with algebra). The term '(conductivity / volumetric heat
capacity)' is used often enough that it is given it's own name and often is
listed separately in tables for materials as 'thermal diffusivity'.
To find out how much the temperature will rise at some distance dX to rise
over a given period, takes a lot of calculus for even a 'simple, flat
plate'. Most calculations require you to convert the various values to
dimensionless numbers and a lot of higher math.
But, if you're handy with Excel or a similar spreadsheet, you can figure out
a very good approximation by breaking down the plate into a whole lot of
thin strips running perpendicular to the heat flow. Suppose you chose to
break it down into 10 strips. Set up the first column of the spreadsheet
with a series of numbers from 0 to some arbitrarily large number of seconds
(say, count from 0 to 500 downward). In the next column put the temperature
of the first strip at time zero (say, 100C). Skip a column for the moment
and in the fourth column put the temperature of the second strip at time
zero (say, 20C). Now, back to the third column put in a formula to
calculate the heat transfer between the two strips based on these two
temperatures (something like '=<conductivity>*<area>*(b1-d1)/<width> ').
Now, take the third and forth column and copy them to the fifth and sixth.
Repeat copying like this until you have a total of ten copies across.
Copy all but the first column down to the row right underneath it. But now
you have to *change* all the temperature columns except the first one. The
new temperature at time '1' for each strip is the previous temperature, plus
the temperature rise caused by the heat addition (for d2, something like
'=d1+c1*<deltatime>/(<heat capacity>*<area>*<width>)' ).
Now, take the second row and copy it down as many times as you want. You
should see the second strip's temperature rise first, followed by the other
strips as the heat travels across the strips in a sort of 'bow wave' affect.
To answer your original question, follow the temperature of the last 'strip'
and decide at what temperature it will be 'too hot to hold', and simply look
across to the left to see what time step this temperature is reached.
Sometimes this sort of 'explicit integration' can be much easier than all
the calculus that would normally be required. If you follow Nick's posts
often, you'll recognize this is a technique he often uses with VB to solve
transient problems. I find for 'one-of' types of calcs, it is easy to just
set up several columns in Excel and then copy/paste for as many iterations
(each row as a time-step) as needed. Of course, the finer the time step,
and the thinner the 'strips' are (although you'll need more columns to
simulate all the heat flows and temperatures in more steps), the more
accurate the answer.
A good text on transient heat flow is "Principles of Heat Transfer 3rd
Edition" by Kreith. Although a lot of it uses differential equations and
calculus, some of the simple cases such as flat plate can be solved with the
use of a couple of charts and converting the problem to 'dimensionless
numbers and ratios'.
Yes, the texts I have on the subject also discuss periodic variations in the
applied heat and calculating the temperature through the material T(t,x) as
a quite complicated function. But in some simple cases, such as starting
with a uniform temperature, and applying the heat as a step change from zero
to some value at t=0, it simplifies quite a bit.
Yes, before the advant of computers for solving this sort of problem in
simple step-wise integration, quite a lot of the work was done in calculus
and Fourier analysis. And a few assumptions to eliminate higher order
effects was common. But hey, we got to the moon and built the atom-bomb
using slide-rules, so it worked.
The only 'trick' to using iterative numerical methods is knowing when to
throw away the answers ;-)
Thanks to everyone. I think I have an adequate idea of the problem now.
I'm getting old and the brain ain't what it was. It looked simple and I was
bothered that I couldn't figure it out myself. It is now evident that it
isn't that simple so I don't feel so bad.
I do use Excel quite a lot and I will try the technique suggested by
I want to be able to estimate how the temperature and pressure of a heated
and confined gas will vary with time when being heated (or cooled) by water
inside a heat exchanger finned pipe. Considering surface area,
temperatures, specific heat and gravity etc for the air/metal interface is
easy but adding the mass and volume of the metal in the heat exchanger slows
things down a lot. Adding the time it will take for the heat to travel from
the water to the ends of the fins makes it even slower. Tough old life is
it not? It does help keep the old brain functioning though.
Thanks again to all who replied. I'll stop bothering you, for a while at